values Eigen values of the common factor solution
e.valuesEigen values of the original matrix
communalityCommunality estimates for each item. These are merely the sum of squared factor loadings for that item.
communalitiesIf using minrank factor analysis, these are the communalities reflecting the total amount of common variance. They will exceed the communality (above) which is the model estimated common variance.
rotationwhich rotation was requested?
loadingsAn item by factor (pattern) loading matrix of class ``loadings" Suitable for use in other programs (e.g., GPA rotation or factor2cluster. To show these by sorted order, use print.psych with sort=TRUE
complexityHoffman's index of complexity for each item. This is just \(\frac{(\Sigma a_i^2)^2}{\Sigma a_i^4}\) where a_i is the factor loading on the ith factor. From Hofmann (1978), MBR. See also Pettersson and Turkheimer (2010).
StructureAn item by factor structure matrix of class ``loadings". This is just the loadings (pattern) matrix times the factor intercorrelation matrix.
fitHow well does the factor model reproduce the correlation matrix. This is just \(\frac{\Sigma r_{ij}^2 - \Sigma r^{*2}_{ij} }{\Sigma r_{ij}^2}
\) (See VSS, ICLUST, and principal for this fit statistic.
fit.offhow well are the off diagonal elements reproduced?
dofDegrees of Freedom for this model. This is the number of observed correlations minus the number of independent parameters. Let n=Number of items, nf = number of factors then
\(dof = n * (n-1)/2 - n * nf + nf*(nf-1)/2\)
objectiveValue of the function that is minimized by a maximum likelihood procedures. This is reported for comparison purposes and as a way to estimate chi square goodness of fit. The objective function is
\(f = log(trace ((FF'+U2)^{-1} R) - log(|(FF'+U2)^{-1} R|) - n.items\). When using MLE, this function is minimized. When using OLS (minres), although we are not minimizing this function directly, we can still calculate it in order to compare the solution to a MLE fit.
STATISTICIf the number of observations is specified or found, this is a chi square based upon the objective function, f (see above). Using the formula from factanal(which seems to be Bartlett's test) :
\(\chi^2 = (n.obs - 1 - (2 * p + 5)/6 - (2 * factors)/3)) * f \)
PVALIf n.obs > 0, then what is the probability of observing a chisquare this large or larger?
PhiIf oblique rotations (using oblimin from the GPArotation package or promax) are requested, what is the interfactor correlation.
communality.iterationsThe history of the communality estimates (For principal axis only.) Probably only useful for teaching what happens in the process of iterative fitting.
residualThe matrix of residual correlations after the factor model is applied. To display it conveniently, use the residuals command.
chiWhen normal theory fails (e.g., in the case of non-positive definite matrices), it useful to examine the empirically derived \(\chi^2\) based upon the sum of the squared residuals * N. This will differ slightly from the MLE estimate which is based upon the fitting function rather than the actual residuals.
rmsThis is the sum of the squared (off diagonal residuals) divided by the degrees of freedom. Comparable to an RMSEA which, because it is based upon \(\chi^2\), requires the number of observations to be specified. The rms is an empirical value while the RMSEA is based upon normal theory and the non-central \(\chi^2\) distribution. That is to say, if the residuals are particularly non-normal, the rms value and the associated \(\chi^2\) and RMSEA can differ substantially.
crmsrms adjusted for degrees of freedom
RMSEAThe Root Mean Square Error of Approximation is based upon the non-central
\(\chi^2\) distribution and the \(\chi^2\) estimate found from the MLE fitting function. With normal theory data, this is fine. But when the residuals are not distributed according to a noncentral \(\chi^2\), this can give very strange values. (And thus the confidence intervals can not be calculated.) The RMSEA is a conventional index of goodness (badness) of fit but it is also useful to examine the actual rms values.
TLIThe Tucker Lewis Index of factoring reliability which is also known as the non-normed fit index.
BICBased upon \(\chi^2\) with the assumption of normal theory and using the \(\chi^2\) found using the objective function defined above. This is just \(\chi^2 - 2 df\)
eBICWhen normal theory fails (e.g., in the case of non-positive definite matrices), it useful to examine the empirically derived eBIC based upon the empirical \(\chi^2\) - 2 df.
R2The multiple R square between the factors and factor score estimates, if they were to be found. (From Grice, 2001). Derived from R2 is is the minimum correlation between any two factor estimates = 2R2-1.
r.scoresThe correlations of the factor score estimates using the specified model, if they were to be found. Comparing these correlations with that of the scores themselves will show, if an alternative estimate of factor scores is used (e.g., the tenBerge method), the problem of factor indeterminacy. For these correlations will not necessarily be the same.
weightsThe beta weights to find the factor score estimates. These are also used by the predict.psych function to find predicted factor scores for new cases.
scoresThe factor scores as requested. Note that these scores reflect the choice of the way scores should be estimated (see scores in the input). That is, simple regression ("Thurstone"), correlaton preserving ("tenBerge") as well as "Anderson" and "Bartlett" using the appropriate algorithms (see factor.scores). The correlation between factor score estimates (r.scores) is based upon using the regression/Thurstone approach. The actual correlation between scores will reflect the rotation algorithm chosen and may be found by correlating those scores.
validThe validity coffiecient of course coded (unit weighted) factor score estimates (From Grice, 2001)
score.corThe correlation matrix of course coded (unit weighted) factor score estimates, if they were to be found, based upon the loadings matrix rather than the weights matrix.
rot.matThe rotation matrix as returned from GPArotation.